In a very different context, devoted to the mathematical understanding of the phenomenon of convergence of internal [1]} and inertial [2]} waves towards attractors, as observed experimentally in linearly stratified flows [3]}, [4]}, [5]}, the authors of Ref. [6]} propose an original interpretation. They show that these waves, whose dispersion relation between their wavelength and their frequency is very peculiar, can be obtained as solutions of a linear partial differential equation (PDE), supplemented by an external forcing, where enters an homogeneous operator of degree 0 [7]}, [8]}. Furthermore, and as a consequence of the nature of this operator, the phenomenon of attraction of waves is seen as a cascading process [6]}. As we will see, this operator can be interpreted as a linear transport in the Fourier space, and interestingly, its discretized version coincides with the linear shell model developed in Ref. [10]}. It thus becomes very tempting to include such an operator in a dynamics that would transfer energy from the large to the small scales, as demanded by the phenomenology of turbulence. Doing so, this would mean that this cascading process could be captured by a linear mechanism. This is what we propose to study in the present article.
